Perspective and ellipses
Ellipses on the surface of a picture
Sherief Hammad, John M. Kennedy, Igor Juricevic and Shazma Rajani
University of Toronto at Scarborough
RUNNING HEAD: Perspective and ellipses
Correspondence to: John M. Kennedy
University of Toronto at Scarborough
1265 Military Trail
Toronto ON M1C1A4, Canada
Tel: 415-287-7435
Fax: 416-287-7676
Email: [email protected]
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Perspective and ellipses
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Abstract
Shapes on the surface of a perspective picture may be misperceived. Subjects
picked a match for an ellipse depicting the circular top of a cylinder. The top was
depicted as tilted from 50 to 850 around a horizontal axis, generating a series of ellipses
on the picture surface. The matches were biased towards a circle over a wide range of
midrange tilts, which suggests they were seen as in-between the shape on the surface and
the shape they depicted.
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Ellipses on a picture’s surface
When we are drawing, we often begin by outlining an area on the picture surface
to match a shape in the scene. Of interest here is that the area on the surface will often
change its appearance as soon as a perspective pattern is added to show the depth in the
scene (Juricevic and Kennedy, 2006; Kennedy, Juricevic, Hammad and Rajani, in press).
In the present study, we consider an elliptical area on the picture surface and how
it looks when it depicts the top of a tilted cylinder drawn in perspective. We ask subjects
to pick the correct match for the ellipse from a set of ellipses (Figures 1 and 2). We ask
whether the match is biased towards the ratio of a circle, as several theories predict, and
how errors are related to the cylinder’s tilt. Does bias arise at extreme tilts of 50 and 850,
at moderate tilts of 450, or across the range of tilt angles?
----------------------------------------------Insert Figures 1 and 2 about here
----------------------------------------------Perspective and a picture surface
Geometrical illusions in which shapes on surfaces are reliably misperceived may
often be due to perspective and mis-applied size constancy (Gregory, 1972; 1980). They
have been of interest since the development of linear perspective in the 1400’s and have
been debated at length in Gestalt and ecological theory of pictures (Arnheim, 1954;
Gibson, 1979).
To focus attention on the topic of interest it may be helpful to consider two
examples, one to do with angles and the other with “table tops”.
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May observers misperceive angles in Figure 3, which is based on a sketch in
Kennedy & Juricevic (2006). In Figure 3, two perspective drawings of cubes have
quadrilateral tops. In the nine quadrilaterals below are the matches for the tops. Most
observers are surprised to learn that the match for the top of the left cube drawing is row
3, column 3 and for the right cube is row 2, column 2. In the cube drawing on the left the
obtuse top-most internal angle is 1430. Many observers suggest its closest match is found
at row 1 column 1, though that is 1300. Matching its acute angle (410) with row 1,
column 1’s acute angle (540) is an overestimation of 130, almost one-third in error.
---------------------------------------------Insert Figure 3 about here
---------------------------------------------An effect of perspective on shape perception occurs in the Shepard (1990) “table
tops” illusion. Table-top shapes are congruent on the picture surface but do not look it.
One is foreshortened along its length and the other across its width. What is notable for
the present purposes is that the foreshortening not only changes the dimensions of the
depicted table-tops, it affects the apparent sizes of the shapes on the picture surface.
Projection from a 3-D scene to a 2-D surface
In Figure 1, cylinders are drawn on a 2D surface with foreshortening of the
circular top and the lines representing the height of the cylinder indicating the cylinder is
tilted. The cylinders are drawn in one-point “linear” projection. The top and the bottom of
each cylinder project as ellipses. The left and right sides of each drawing converge
towards the bottom ellipse. The convergence indicates tilt (see Figure 4).
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-------------------------------------Insert Figure 4 here
-------------------------------------Consider a highly extreme ellipse in Figure 1. The information it supplies directly
is about its own aspect ratio. That is, the oval lines in a picture of a cylinder allow
“direct” perception of their own aspect ratio, and in addition the “indirect” perception of
circular tops and bottoms of a depicted object (Gibson, 1979; Arnheim, 1954). We test
direct perception of the narrow ellipse on the picture surface to discover how it is affected
by the lines allowing the indirect perception of the circular form.
We use a matching task here to avoid subjects having to report ratios as numbers,
and to permit subjects to report that two forms simply look alike.
Projective, Classification, Good-form and Perspective-features theories
Arnheim (1954) and Sedgwick & Nicholls (1993) suggest “crosstalk” between the
optic information specifying a depicted form and the form’s projection on a picture
surface gives rise to “in-between” percepts that are neither veridical about the picture
surface area nor full-constancy impressions of the depicted object (Sedgwick, 2003).
Crosstalk occurs from the pictured world to the picture surface and vice versa. So far as
judging a projected form is concerned, where might crosstalk be concentrated -- at
extreme tilts, medium tilts or across the range?
A “projective” hypothesis about crosstalk is that as the projections on the surface
deviate from the depicted form, bias increases. That is, tilts of 5 and 15 degrees, at which
cylinder tops project slim ellipses, entail large illusions, middling tilts (45 degrees) less
and at 75 and 85 degrees there is very little crosstalk since the cylinder tops are close to
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parallel to the picture surface and almost project their true forms. In this theory, only
extreme foreshortening triggers strong perceptual corrections.
Alternatively, it may be precisely the cases in which ellipses approach the top’s
form that boost crosstalk. Perhaps in these conditions the ellipse can appear perfectly
circular. In a “classification” theory, top-down effects are said to arise from categorizing
stimuli as showing objects. Close to an aspect ratio of 1, an ellipse categorized as
showing a cylinder’s circular top might be seen as a perfect circle. Categorization effects
might be minimal for aspect ratios far from 1, this theory could contend, since they depart
too much from the shape of the familiar object to be made into perfect circles.
A “good form” theory might predict substantial errors not only for ellipses close
to circular but also for slim ellipses. If physical conditions approach a “simpler” shape, a
percept with full symmetry and equality of its parts is favoured, Gestaltists argue. When
the differences from a good form are modest, “projective distortions are observed only by
the select few who have been trained to watch out for them” (Arnheim, 1986). A possible
implication of “good form” theory is that a very slim ellipse might be seen as close to a
straight line, since a straight line is another good form. That is, very slim ellipses might
be seen as even slimmer than is correct. In this “good-form Gestalt” theory ellipses
projected by both large (e.g. 85 degrees) and small (e.g. 5 degrees) tilt angles will suffer
the most correction. Intermediate angles (45 degrees) will incur less.
A perspective-features theory could argue that at tilts close to 45 degrees the lines
on the page showing convergence and foreshortening are most evident. The lines for the
cylinder’s sides are long enough to converge distinctly at intermediate angles. At 5 and
15 degrees tilt, the lines are long but their convergence is minimal. At a tilt close to 85
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degrees, convergence is present but the lines are too short and stubby for it to be obvious.
If so, lines on the page with perspective features most readily offer crosstalk at
intermediate tilts.
In sum, to test 4 hypotheses about the tilts creating errors, we ask subjects to
match ellipses in a set of drawings of a tilted cylinder with ellipses in a set of ellipses.
Method
Participants
Eleven participants (8 female, mean age 19, range 4 years) were recruited from a
first year psychology class at the University of Toronto at Scarborough. Each had normal
or corrected-to-normal vision and was naïve to the purpose of the experiment.
Stimuli
All stimuli were generated in a bespoke program called Angle Grinder, written in
Java programming language and using the “Java 3D API” to render the cylinders. They
were presented to all participants on a 15” computer monitor. Cylinders were drawn in
linear perspective with white lines 1mm thick on a black background. They were
designed as “wire-frame” presentations, rather than “solid” opaque objects, showing the
upper-most and lower-most ellipses, as well as the two upright lines depicting the body of
the cylinder.
The top face was tilted from the horizontal in 10º increments between 5º and 85º.
At 5º the ellipses were highly foreshortened. At 85º the projections of the ellipses were
very close to a circle. Each tilt was a rotation out of the picture plane about the nearest
point of the upper-most ellipse, which was fixed at eye level to the observer. At a 90º tilt
the top would have been projected as a straight line. As a result of this manipulation the
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height of the upper-most ellipse (as measured between the lowest and highest points)
varied from 3mm (5º tilt) to 34mm (85º tilt).
Each participant was shown the 9 cylinder drawings ten times in a random order,
with no two participants receiving the same order, in two blocks; in one, the cylinder was
present, and the other had only the cylinder’s upper-most ellipse (Figure 5). Half of the
subjects were presented with the cylinder-drawings first, and half were given the ellipsesalone first.
------------------------------------------Insert Figure 5 about here
-------------- --------------------------Procedure
Participants sat approximately 50cm from the computer monitor. In one order,
they were first told that they would be presented with ellipses of different sizes on the
computer screen (ellipse-only condition), and were shown examples of ellipses taken
from actual stimuli (i.e., the tops of the depicted cylinders). They were to use the mouse
pointer to click on the matching ellipse at the lower half of the screen. The order of
ellipses on the lower half of the screen was randomized with each presentation.
After 10 presentations of each ellipse (90 trials), instructions were given for the
second block (also 90 trials). In this, participants were informed that they would be
presented with two ellipses, one above the other and connected at the sides by two lines
(cylinder condition). They were instructed to use the mouse pointer to match the uppermost ellipse at the top of the screen with the corresponding ellipse at the lower half of the
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screen. As noted above, half the participants received the ellipse-only condition first, and
half received the cylinder condition first.
It is important to note that the instructions ask subjects to match the ellipses,
not the cylinder that is depicted, in describing the stimuli. The depicted cylinder has a
circular top, no matter what the tilt. The ellipse on the screen varies from trial to trial. The
instructions ran as follows: “You will now be presented with two ellipses on the
computer screen, joined at their edges by two straight lines. The ellipses you see will be
of different sizes. Some may be short and some may be tall. In some cases the two
ellipses may overlap.” The aim was to have participants to judge the upper-most 2D
ellipses that vary in shape, rather than the constant 3D object depicted in the display, and
to match the ellipse that is present on a given trial to the corresponding shape at the lower
half of the screen.
Results
The height of each ellipse presented, subtracted from the participants mean
estimations, results in mm values about 0. Overestimations are positive. The mean
responses and errors for both conditions are represented in Figures 7 and 8. For
significance, an alpha level of 0.05 was applied to all comparisons.
----------------------------------------Insert Figures 7 and 8 about here
----------------------------------------Cylinder-top ellipses were larger than ellipses-alone on a 2 (ellipse vs. cylinder)
X 9 (presented ellipse height mm: 3, 8, 13, 18, 22, 26, 30, 32, 34) within-subjects
ANOVA, F(1, 8) = 29.3, MSE = 119.2. Cylinder-top ellipses were judged different from
Perspective and ellipses
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ellipses-alone only at midrange tilt angles (main effect F(8, 64) = 29.2, MSE = 45.0;
paired-sample T-tests different at angles of: 25º/13mm (t(10) = 4.53), 35º/18mm (t(10) =
5.79), 45º/22mm (t(10) = 6.50), 55º/26mm (t(10) = 3.66). Hence there was a significant
interaction between cylinder present or absent and tilt angle (F(8, 64) = 17.2, MSE =
21.9).
Cylinder top-faces were judged greater than ellipses on their own, with
differences ranging from -1.9mm (at 85º tilt) to 4.6mm (at 25º tilt), a total of 6.5mm. For
ellipses-alone, the range is small, between -1.3mm (at 85º tilt) and 1.1mm (at 25º tilt), a
total of 2.4mm.
In sum, ellipses depicting cylinder tops are judged larger than ellipses on their
own at intermediate tilts.
Discussion
Errors were most evident from 25º to 55º. The errors are overestimations. A
theory about perspective’s features is supported rather than projection, good-form or
classification hypotheses.
Contrary to the projection hypothesis, the largest errors were not for the slimmest
ellipses. Good-form Gestalt theory fails since the biases were not most evident at extreme
tilts. Classification theory expected the largest errors to occur as ellipses approach the
circular form of the depicted object, but they were at intermediate angles.
Since the test and matched ellipses were presented simultaneously, it seems
unlikely that subjects forgot the ellipse’s form and then made judgements on the basis on
the depicted form. Indeed, if they had forgotten, all the judgments would have been of
circular forms. (We suggest that errors might grow if the time between exposure and
Perspective and ellipses
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matching was expanded and distracting tasks intervened, but memory effects like these
are subsequent to perception, our primary focus.)
Errors concentrated at middling values support the perspective-features
hypothesis. Convergence and foreshortening are shown most clearly on the picture
surface at these intermediate angles, whereas tilts close to 0º and 90º incur short lines or
minimum convergence.
Perspective has many consequences for perception. Physically, it controls the
angles subtended by objects at the observer’s vantage point and the projections onto a
picture surface. Psychologically, it is the geometry that vision must approximate in
dealing with many constancy problems, and needs to deal with in looking at pictorial
surfaces. The present study indicates – in terms from Gregory (1972) – that vision
misapplies perspective features to the forms it projects onto pictorial flat surfaces. On the
one hand, vision is using perspective in looking at surfaces of any kind, and on the other
hand prominent perspective features drawn on the surface bias observers judging features
of the surface itself.
Logically, in normal environmental circumstances perspective may apply even
handedly, without bias. This would occur if converging lines on textured surfaces were
evenly distributed. But a textured surface projects a gradient to the observer’s vantage
point. That is, the evenly-distributed texture on the surface projects optic texture that has
a bias. Vision’s task is to use the bias in the gradient to have the observer see the slant of
the surface and the true form of parts of the surface. In pictures, the gradient is presented
on a surface. That is, the markings on the surface contain a bias – features to do with
perspective. Vision’s task is now to discern the true shape despite the bias. The results of
Perspective and ellipses
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our experiment suggest the task of discounting the bias from perspective features is not
solved perfectly.
Our results suggest as many questions as they answer. For example, if the
cylinder was transformed into a truncated cone, with the top ellipse at the high, narrow
end of the cone, convergence would be more obvious at lower tilts than for cylinders. In
contrast, it would be most obvious for higher tilts if the top was at the cone’s broader end.
If the cones were combined like some beer glasses that have a slim waist, the
convergence of the lines connecting to the top would be different from the lines
connecting to the bottom ellipse. This shape could help distinguish information for tilt
from effects simply due to converging lines. Indeed, perhaps the trickiest issues here are
to do with the relation between high-level perspective information, which is about depth
and pictured shape, and flat features which may produce low-level pattern effects in
vision irrespective of the high-level information. We have used tilt information here in
the present study, and the effects are consistent with moderate crosstalk from
foreshortening effects, but in principle it is hard to distinguish possible foreshortening
from effects from the converging lines that carry information about foreshortening.
However, one advantage of the theory of misattribution of perspective is that it indicates
what geometrical variables to explore systematically – tilt, foreshortening, convergence
and other aspects of perspective. If these have crosstalk effects in accord with the spatial
depth and slant they signify then the perspective theory is useful indeed.
Figures 1 and 3 depict highly familiar symmetrical forms. The perspectivefeatures theory of crosstalk should apply broadly, for example to unfamiliar and
asymmetrical forms. Figure 1 allows observations to be biased towards a circle, and the
Perspective and ellipses
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bias for Figure 3 is a square. But other forms could be inscribed within these e.g. a
scalene or isosceles triangle, or an irregular pentagon. If the perspective-features theory is
correct, the angles that subjects perceive the depicted referents having should be the
angles to which the shape on the surface appears biased.
In conclusion, observers’ perceptions of ellipses on the surface of a perspective
picture are biased towards the referent of the ellipses.
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References
Arnheim, R. (1954) Art and visual perception. University of California Press, Berkeley.
Arnheim, R. (1986). New essays on the psychology of art. University of California
Press, Berkeley
Gibson, J. J. (1979) The ecological approach to visual perception. Houghton-Mifflin,
Boston.
Gregory, R. L. (1972) Eye and brain. (2nd Ed) World University Library, London.
Gregory, R. L. (1980) Perceptions as hypotheses. Philosophical Transactions of the Royal
Society of London, Series B, 290, 181-197.
Juricevic, I., & Kennedy, J. M. (2006). Looking at perspective pictures from too far, too
close, and just right. Journal of Experimental Psychology: General, 135, 448-461.
Kennedy, J. M. & Juricevic, I. (2006). Foreshortening, convergence and drawings
from a blind adult. Perception, 35, 847 – 851
Kennedy, J. M., Juricevic, I., Hammad, S. & Rajani, S. (in press) Perspective: Arnheim
and in-between solutions. Psychology of Aesthetics, Creativity and the Arts
Kubovy, M. (1986) The psychology of perspective and Renaissance art. Cambridge
University Press, Cambridge.
Sedgwick, H. A. (2003). Relating direct and indirect perception of spatial layout’. In:
Looking into pictures; an interdisciplinary approach to pictorial space. In Hecht,
H., Schwartz, R, & Atherton, M. (Eds.), Looking into pictures: an
interdisciplinary approach to pictorial space (pp. 239-299). Cambridge, MA: MIT
Press
Sedgwick, H. A. & Nicholls, A. L. (1993). Cross talk between the picture surface and the
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pictured scene: Effects on perceived shape. Perception, 22 (supplement), 109.
Shepard, R. N. (1990). Mind sights: original visual illusions, ambiguities, and other
anomalies, with a commentary on the play of mind in perception and art. New
York: Freeman.
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Acknowledgments
We thank high-school science teachers Seth Bernstein and Joe Hockin (members of
TSTOP, Ontario, 2006) for comments on theories of the angle illusion. We also offer our
appreciation to Sandacre Technology for their programming expertise.
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Figure Captions
Figure 1. Orientation of cylinder stimuli through nine tilt angles from the horizontal.
Figure 2. Test condition with a cylinder depicted, and 9 ellipses to match to the
cylinder’s top ellipse. Many observers match the cylinder top with the ellipse in the upper
row, third from the left, though the correct match is upper row, leftmost ellipse.
Figure 3. Two of the quadrilaterals correspond to the top faces of the two cubes, but
observers misidentify the matches. The correct matches are the middle and bottom-right
quadrilaterals.
Figure 4. The extended lines converge in one-point perspective, indicating that the
cylinder is tilted (450 from horizontal).
Figure 5. Test condition with an ellipse at the top and a set of 9 ellipses below
Perspective and ellipses
Figure 1.
50/150/250
350/450/550
650/750/850
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Perspective and ellipses
Figure 2.
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Perspective and ellipses
Figure 3.
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Figure 4.
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Figure 5.
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Figure 6. Mean heights of ellipse chosen at each tilt.
Height presented mm
35
30
25
20
ellipses
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cylinder top-faces
10
5
0
3(5)
8(15) 13(25) 18(35) 22(45) 26(55) 30(65) 32(75) 34(85)
Height presented mm (degree of tilt from
horizontal)
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Figure 7. Mean errors at each tilt.
Errors in heights chosen mm
5
4
3
2
ellipses
1
cylinder top-faces
0
-1
3(5)
8(15) 13(25) 18(35) 22(45) 26(55) 30(65) 32(75) 34(85)
-2
-3
Height presented mm (degree of tilt from
horizontal)